Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks

Details Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks Level Crossing Probabilities II: Polygonal Recurrence of Multidimensional Random Walks

2004-06-22

arxiv.org/abs/math/0406423v3

Mathematics

Probability

23 pages, errors and typos corrected, references added

Scientific paper

In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the sense of Mauldin, Monticino and v.Weizsaecker [1] is transient. On the other hand we construct an example of a transient but polygonally recurrent random walk with independent components on Z^2.

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